3.386 \(\int \sqrt{\frac{-a+b x}{x^2}} \, dx\)

Optimal. Leaf size=53 \[ 2 x \sqrt{\frac{b}{x}-\frac{a}{x^2}}+2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{\frac{b}{x}-\frac{a}{x^2}}}\right ) \]

[Out]

2*Sqrt[-(a/x^2) + b/x]*x + 2*Sqrt[a]*ArcTan[Sqrt[a]/(Sqrt[-(a/x^2) + b/x]*x)]

________________________________________________________________________________________

Rubi [A]  time = 0.0789457, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1979, 2007, 2013, 620, 203} \[ 2 x \sqrt{\frac{b}{x}-\frac{a}{x^2}}+2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{\frac{b}{x}-\frac{a}{x^2}}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[(-a + b*x)/x^2],x]

[Out]

2*Sqrt[-(a/x^2) + b/x]*x + 2*Sqrt[a]*ArcTan[Sqrt[a]/(Sqrt[-(a/x^2) + b/x]*x)]

Rule 1979

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && GeneralizedBinomialQ[u, x] &&  !Gene
ralizedBinomialMatchQ[u, x]

Rule 2007

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(x*(a*x^j + b*x^n)^p)/(p*(n - j)), x] + Dist
[a, Int[x^j*(a*x^j + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, j, n}, x] && IGtQ[p + 1/2, 0] && NeQ[n, j] && EqQ[
Simplify[j*p + 1], 0]

Rule 2013

Int[(x_)^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[(a*x^Simplify[j/n]
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && IntegerQ[Simplify[j
/n]] && EqQ[Simplify[m - n + 1], 0]

Rule 620

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \sqrt{\frac{-a+b x}{x^2}} \, dx &=\int \sqrt{-\frac{a}{x^2}+\frac{b}{x}} \, dx\\ &=2 \sqrt{-\frac{a}{x^2}+\frac{b}{x}} x-a \int \frac{1}{\sqrt{-\frac{a}{x^2}+\frac{b}{x}} x^2} \, dx\\ &=2 \sqrt{-\frac{a}{x^2}+\frac{b}{x}} x+a \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x-a x^2}} \, dx,x,\frac{1}{x}\right )\\ &=2 \sqrt{-\frac{a}{x^2}+\frac{b}{x}} x+(2 a) \operatorname{Subst}\left (\int \frac{1}{1+a x^2} \, dx,x,\frac{1}{\sqrt{-\frac{a}{x^2}+\frac{b}{x}} x}\right )\\ &=2 \sqrt{-\frac{a}{x^2}+\frac{b}{x}} x+2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a}}{\sqrt{-\frac{a}{x^2}+\frac{b}{x}} x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0295023, size = 66, normalized size = 1.25 \[ \frac{2 x \sqrt{\frac{b x-a}{x^2}} \left (\sqrt{b x-a}-\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )\right )}{\sqrt{b x-a}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[(-a + b*x)/x^2],x]

[Out]

(2*x*Sqrt[(-a + b*x)/x^2]*(Sqrt[-a + b*x] - Sqrt[a]*ArcTan[Sqrt[-a + b*x]/Sqrt[a]]))/Sqrt[-a + b*x]

________________________________________________________________________________________

Maple [A]  time = 0.01, size = 56, normalized size = 1.1 \begin{align*} -2\,{\frac{x}{\sqrt{bx-a}}\sqrt{{\frac{bx-a}{{x}^{2}}}} \left ( \sqrt{a}\arctan \left ({\frac{\sqrt{bx-a}}{\sqrt{a}}} \right ) -\sqrt{bx-a} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x-a)/x^2)^(1/2),x)

[Out]

-2*((b*x-a)/x^2)^(1/2)*x*(a^(1/2)*arctan((b*x-a)^(1/2)/a^(1/2))-(b*x-a)^(1/2))/(b*x-a)^(1/2)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\frac{b x - a}{x^{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x-a)/x^2)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt((b*x - a)/x^2), x)

________________________________________________________________________________________

Fricas [A]  time = 0.857931, size = 228, normalized size = 4.3 \begin{align*} \left [2 \, x \sqrt{\frac{b x - a}{x^{2}}} + \sqrt{-a} \log \left (\frac{b x - 2 \, \sqrt{-a} x \sqrt{\frac{b x - a}{x^{2}}} - 2 \, a}{x}\right ), 2 \, x \sqrt{\frac{b x - a}{x^{2}}} - 2 \, \sqrt{a} \arctan \left (\frac{x \sqrt{\frac{b x - a}{x^{2}}}}{\sqrt{a}}\right )\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x-a)/x^2)^(1/2),x, algorithm="fricas")

[Out]

[2*x*sqrt((b*x - a)/x^2) + sqrt(-a)*log((b*x - 2*sqrt(-a)*x*sqrt((b*x - a)/x^2) - 2*a)/x), 2*x*sqrt((b*x - a)/
x^2) - 2*sqrt(a)*arctan(x*sqrt((b*x - a)/x^2)/sqrt(a))]

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\frac{- a + b x}{x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x-a)/x**2)**(1/2),x)

[Out]

Integral(sqrt((-a + b*x)/x**2), x)

________________________________________________________________________________________

Giac [A]  time = 1.28611, size = 82, normalized size = 1.55 \begin{align*} -2 \,{\left (\sqrt{a} \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right ) - \sqrt{b x - a}\right )} \mathrm{sgn}\left (x\right ) + 2 \,{\left (\sqrt{a} \arctan \left (\frac{\sqrt{-a}}{\sqrt{a}}\right ) - \sqrt{-a}\right )} \mathrm{sgn}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x-a)/x^2)^(1/2),x, algorithm="giac")

[Out]

-2*(sqrt(a)*arctan(sqrt(b*x - a)/sqrt(a)) - sqrt(b*x - a))*sgn(x) + 2*(sqrt(a)*arctan(sqrt(-a)/sqrt(a)) - sqrt
(-a))*sgn(x)